Page 171 - Analysis and Design of Energy Geostructures

144 Analysis and Design of Energy Geostructures

1
ð4:5Þ
3
ε ij 5 ε v δ ij 1 e ij
where ε v is a scalar quantity called volumetric strain, δ ij is the Kronecker delta (which
is equal to 1 if i 5 j and to 0 otherwise) and e ij is a tensor characterised by zero trace
called deviatoric strain tensor (or strain deviator). The volumetric strain can be written
in rectangular Cartesian coordinates as

ð4:6Þ
ε v 5 ε kk 5 tr ε ij 5 ε xx 1 ε yy 1 ε zz
The deviatoric strain tensor in rectangular Cartesian coordinates reads

2 3
e xx
e ij 5 4 ε yx ε xy ε xz 5 ð4:7Þ
e yy
ε yz
ε zy
e zz
ε zx
where e ii 5 ε ii 2 ε v =3.
Eq. (4.1) can therefore be rewritten in matrix form as
2 3 2 3 2 3
0 0
1 ε v e xx ε xy ε xz
ε xy
ε xz
ε xx
5 1
4 5 4 0 0 5 4 5 ð4:8Þ
3 ε v ε yx e yy ε yz
ε yz
ε yx
ε yy
0 0 ε v ε zx ε zy e zz
ε zx
ε zy
ε zz
4.3.4 Principal strains
A feature of the strain tensor, similar to any symmetric tensor, is that at every material
point of any coordinate system there exist three mutually perpendicular planes, called
principal planes, along which zero shear strains are observed. The normal strains on
these planes are called principal strains. The principal strains and the associated princi-
pal directions can be written as the eigenvalues and the eigenvectors of the strain ten-
sor, respectively. Invariants can be defined for the strain tensor.
4.4 Compatibility equations

The concept of strain compatibility refers to the physical concept that when deforma-
tion occurs in a continuum material it happens without material gaps or overlaps. This
means that the relation between material points before and after the variation of con-
figuration remains the same (cf. Fig. 4.4). When displacements are known, strains can
be calculated according to expressions (4.2) (4.4) by computing partial derivatives.
However, the inverse problem of determining displacements from strains, which is
usually encountered in stress-based formulations, is overdetermined because either of
expressions (4.2), (4.3) or (4.4) contains six equations and only three unknowns. In